Harmonic oscillator statistical mechanics

The chapter starts with a brief review of thermodynamic principles as they apply to the concept of the chemical equilibrium. That section is followed by a short review of the use of statistical thermodynamics for the numerical calculation of thermodynamic equilibrium constants in terms of the chemical potential (often designated as (i). Lastly, this statistical mechanical development is applied to the calculation of isotope effects on equilibrium constants, and then extended to treat kinetic isotope effects using the transition state model. These applications will concentrate on equilibrium constants in the ideal gas phase with the molecules considered in the rigid rotor, harmonic oscillator approximation. 

Thus the first correction to the classical statistical mechanics at high temperature goes as h2. There are higher order corrections. The result obtained here is identical to that found by J. Kirkwood for a harmonic oscillator. The approach to the... 

In the classical high-temperature limit, kBT hv, where kB is the Boltzmann constant, and hv is the spacing of the quantum-mechanical harmonic oscillator energy levels. If this condition is fulfilled, the energy levels may be considered as continuous, and Boltzmann statistics apply. The corresponding distribution is... 

For POM, a matrix algorithm for the statistical mechanical treatment of an unperturbed -A-B-A-B- polymer chain with energy correlation between first-neighboring skeletal rotations is described. The results of the unperturbed dimensions are in satisfactory agreement with experimental data. In addition, if the same energy data are used, the results are rather close to those obtained by the RIS scheme usually adopted. The RIS scheme is shown to be also adequate for the calculation of the average intramolecular conformational energy, if the torsional oscillation about skeletal bonds is taken into account in the harmonic approximation. 

Let the total system of harmonic oscillator plus radiation field come to equilibrium. Then we know from equilibrium statistical mechanics... 

The hat-curved-harmonic oscillator model, unlike other descriptions of the complex permittivity available now for us [17, 55, 56, 64], gives some insight into the mechanisms governing the experimental spectra. Namely, the estimated relaxation time of a nonrigid dipole (xovib 0.2 ps) is close to that determined in the course of very accurate experimental investigations and of their statistical treatment [17, 54-56]. The reduced parameters presented in Tables XIVA and XIVB and the form of the hat-curved potential well (determined by the parameters u, (3, f) do not show marked dependence on the temperature, while the spectra themselves vary with T in greater extent. We shall continue discussion of these results in Section X.A. 

The statistical mechanical form of transition theory makes considerable use of the concepts of translational, rotational and vibrational degrees of freedom. For a system made up of N atoms, 3N coordinates are required to completely specify the positions of all the atoms. Three coordinates are required to specify translational motion in space. For a linear molecule, two coordinates are required to specify rotation of the molecule as a whole, while for a non-linear molecule three coordinates are required. These correspond to two and three rotational modes respectively. This leaves 3N — 5 (linear) or 3N — 6 (non-linear) coordinates to specify vibrations in the molecule. If the vibrations can be approximated to harmonic oscillators, these will correspond to normal modes of vibration. 

The probability is calculated according to classical statistical mechanics (Appendix A.2). According to Eq. (A.49), the density of states (number of states per unit energy) for s uncoupled harmonic oscillators with frequencies z/j is... 

The procedure, known as second quantization, developed as an essential first step in the formulation of quantum statistical mechanics, which, as in the Boltzmann version, is based on the interaction between particles. In the Schrodinger picture the only particle-like structures are associated with waves in 3N-dimensional configuration space. In the Heisenberg picture particles appear by assumption. Recall, that in order to substantiate the reality of photons, it was necessary to quantize the electromagnetic field as an infinite number of harmonic oscillators. By the same device, quantization of the scalar r/>-field, defined in configuration space, produces an equivalent description of an infinite number of particles in 3-dimensional space [35, 36]. The assumed symmetry of the sub-space in three dimensions decides whether these particles are bosons or fermions. The crucial point is that, with their number indeterminate, the particles cannot be considered individuals [37], but rather as intuitively understandable 3-dimensional waves - (Born) -with a continuous density of energy and momentum - (Heisenberg). 

There are well-known temperature effects, particularly dealing with the two first moments of the spectra that evoke those of the thermal average appearing in the statistical mechanics of quantum harmonic oscillator coordinates. 

Statistical Mechanics of a System of Oscillators.—Dynamically, we have seen that a. crystal can be approximated by a set of ZN vibrations, if there are A afronas in the crystal. These vibrations have fre-quencies which we may label v. . . vzN varying through a, wide range of frequencies. To the approximation to which the restoring forces can be treated as linear, these oscillations are independent of each other, each one corresponding to a simple harmonic oscillation whose frequency is inde-... 

In this example the master equation formalism is appliedto the process of vibrational relaxation of a diatomic molecule represented by a quantum harmonic oscillator In a reduced approach we focus on the dynamics of just this oscillator, and in fact only on its energy. The relaxation described on this level is therefore a particular kind of random walk in the space of the energy levels of this oscillator. It should again be emphasized that this description is constructed in a phenomenological way, and should be regarded as a model. In the construction of such models one tries to build in all available information. In the present case the model relies on quantum mechanics in the weak interaction limit that yields the relevant transition matrix elements between harmonic oscillator levels, and on input from statistical mechanics that imposes a certain condition (detailed balance) on the transition rates. 

Statistical Mechanics of the Harmonic Oscillator. As has already been argued in this chapter, the harmonic oscillator often serves as the basis for the construction of various phenomena in materials. For example, it will serve as the basis for our analysis of vibrations in solids, and, in turn, of our analysis of the vibrational entropy which will be seen to dictate the onset of certain structural phase transformations in solids. We will also see that the harmonic oscillator provides the foundation for consideration of the jumps between adjacent sites that are the microscopic basis of the process of diffusion. 

In light of these claims, it is useful to commence our study of the thermodynamic properties of the harmonic oscillator from the statistical mechanical perspective. In particular, our task is to consider a single harmonic oscillator in presumed contact with a heat reservoir and to seek the various properties of this system, such as its mean energy and specific heat. As we found in the section on quantum mechanics, such an oscillator is characterized by a series of equally spaced energy levels of the form E = (n + )hu>. From the point of view of the previous section on the formalism tied to the canonical distribution, we see that the consideration of this problem amounts to deriving the partition function. In this case it is given by... 

In the text we sketched the treatment of the statistical mechanics of the harmonic oscillator. Flesh out that discussion by explicitly obtaining the partition function, the free energy, the average energy, the entropy and specific heat for such an oscillator. Plot these quantities as a function of temperature. 

What we have learned is that our change to normal coordinates yields a series of independent harmonic oscillators. From the statistical mechanical viewpoint, this signifies that the statistical mechanics of the collective vibrations of the harmonic solid can be evaluated on a mode by mode basis using nothing more than the simple ideas associated with the one-dimensional oscillator that were reviewed in chap. 3. 

Once this result is in hand, the statistical mechanics of this system reduces to repeated application of the statistical mechanics of a simple one-dimensional quantum harmonic oscillator already discussed in chap. 3. This can be seen by noting that the partition function may be written as... 

Here again, therefore, we obtain for our term scheme an equidistant succession of energy levels, as in Bohr s theory. The sole difference lies in the fact that the whole term diagram of quantum mechanics is displaced relative to that of Bohr s theory by half a quantum of energy. Although this difference does not manifest itself in the spectrum, it plays a part in statistical problems. In any case it is important to note that the linear harmonic oscillator possesses energy hv in. the lowest state, the so-called zem-jpoint energy. 

A collection of N harmonic oscillators at thermal equilibrium at absolute temperature T is shown by statistical mechanics to have the thermodynamic energy... 

Only in the high-energy limit does classical statistical mechanics give accurate values for the sum and density of states terms in equation (A3.12.15) [3,14]. Thus, to determine an accurate RRKM k(E) for the general case, quantum statistical mechanics must be used. Since it is difficult to make anharmonic corrections, both the molecule and transition state are often assumed to be a collection of harmonic oscillators for calculating the... 

Thermodynamic properties calculated for the current study are presented in Table 7.5. Enthalpy of formation and entropy values are reported at 298 K, as most experimental data are referenced or available at 298 K. This facilitates the use of these thermodynamic properties and the use of isodesmic reaction set. Entropies and heat capacities are calculated by statistical mechanics using the harmonic-oscillator approximation for vibrations, based on frequencies and moments of inertia of the optimized B3LYP/6-311G(d,p) structures. Torsional frequencies are not included in the contributions to entropy and heat capacities instead, they are replaced with values from a separate analysis on each internal rotor analysis (IR). 

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